In this paper, we propose the new possible texture structures where
there are two texture zeros in \(M_{l}\)  and two cofactor zeros in
\(M_{\nu }\)  (We denote them the matrices with texture/cofactor zeros).
It seems that such mass matrices are rather unusual because one
instinctively expects the type of texture structures to be the same
for both \(M_{l}\)  and \(M_{\nu }\) . However, one reminds the type-I
seesaw model as \( M_{\nu }=-M_{D} M_{R}^{-1}M_{D}^{T}\) . Then the
texture or cofactor zeros of \(M_{\nu }\)  can be attributed to the
texture zeros in \(M_{D}\)  and \(M_{R}\) . Generally, this can be
realized by \(Z_{n}\)  flavor symmetry[1]}, [2]}. Therefore from
the point of flavor symmetry, both texture zeros and cofactor zeros
structures manifest the same flavor symmetry in different ways. It
is our main motivation to carry out this work and a concrete model
will be constructed in the following section. Furthermore, we take
the so-called the parallel \(Ans\ddot{a}tze\)  that the positions of
texture zeros in \(M_{l}\)  are chosen to be the same as the cofactor
zeros in \(M_{\nu }\) . Although there is no priori reason requiring the
parallel structures, they are usually regarded in many literatures
as an esthetical appeal and the precursor of the more general cases.
The lepton mass matrices with parallel texture zero structures have
been systematically investigated in Ref.[3]}. Subsequently,
the idea is generalized to more complicated situations such as
parallel hybrid textures[4]}, parallel cofactor zero
textures[5]}. In our case, there exists \(C^{2}_{6}=15\) 
logically possible patterns for two texture/cofactor zeros in mass
matrices. It is indicated that the 15 textures can be grouped into 4
classes with the matrices in each class connected by \(S_{3}\) 
permutation transformation and sharing the same physical
implications. Among the 4 classes, one of them is not viable
phenomenologically. Therefore we focus on the other three nontrivial
classes.
